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Math · 18 · Problem-Solving and Data Analysis

Descriptive Statistics I

Notes

Key points

  • Mean = sum ÷ count, so sum = mean × count.
  • Median: order the list first; even nn → average the two middles.
  • Mode = most frequent value.
  • Outliers pull the mean; the median barely moves.
  • Right-skewed (long tail high) → mean > median. Left-skewed → mean < median.
  • Add kk to every value → mean and median shift by kk.
  • Frequency table: values × frequencies for the sum; count to the middle for the median.

Formulas

Meanxˉ=sum of valuesn\bar{x}=\dfrac{\text{sum of values}}{n}
Sum from meansum=xˉ⋅n\text{sum}=\bar{x}\cdot n
Median positionn+12\dfrac{n+1}{2}In the ordered list
Combined (weighted) meanxˉ=n1xˉ1+n2xˉ2n1+n2\bar{x}=\dfrac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}

Common traps

  • Finding the median without putting the values in order.
  • Averaging two group means when the groups are different sizes.
  • Reading frequencies in a table as the data values themselves.

SAT tip

Mean questions are sum questions: turn every mean into a total, work with totals, divide at the end.

Test yourself

10 official SAT questions on Descriptive Statistics I, easiest first.

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